I’m surprised only a handful of people have mentioned Ghibli movies. For me it was Nausicaa of the Valley of the Wind, it’s probably one of the first movies I remember watching in general. Still my favorite Ghibli movie and I must have watched it dozens of times as a kid.
The source material for it is a manga by Miyazaki himself and it’s much longer and deeper (the movie only covers about 1.5 out of 7 volumes, and changes a lot of details). Highly recommended.

Princess Mononoke. I randomly picked it up at Blockbuster and it was so different than anything else I’d ever watched. I instantly fell in love with it.

Akira

Sailor Moon. It was on early in the morning and I’d get ready for school really fast so I could sit and watch it before running to catch the bus.

Heidi and Maya the Bee when I was about 4

Death Note, sometime in 2009 or 2010.

I remember when it came out, people on forums kept talking about it nonstop. I had no idea what it was, so I just blindly torrented something called Death Note… it was the live action Japanese adaptation lol. I didn’t even know it was an anime until years later. Just for years I thought everyone on the internet was obsessed with that live action movie.

Cowboy Bebop

Yu Yu Hakushou. I didn’t even realize the name is Yu Yu Hakushou until years later. I only remember thinking the delinquent guy with a laser sword and a guy shooting energy balls from the finger tip’s pretty cool.

If we’re talking about just what you watched before you recognized it was anime, it’s between pokemon, Naruto, and One Piece. Couldn’t tell you which I watched first.

If we’re talking “Oh! I recognize this is anime!” then probably either Deathnote or Fullmetal Alchemist (2003) thanks to mid-2010s Netflix (back when it was still good).

Neon Genesis Evangelism. I had such high hopes…

Very first was probably Inuyasha, Dragonball Z, or some Gundam way back on adult swim as a kid. My first real one that I engaged with as a semi-adult was probably Death Note.

Probably Bee Maja, Wickie and the strong men or Heidi. Tons of classic German children’s shows are actually animes.

First show was probably Voltron. First film was probably Vampire Hunter D.

Toonami became a big part of my life, and there was a small theater downtown that did showings of Miyazaki and such. I remember seeing Metropolis there, too.

I owe a lot to those scrappy little enterprises, taking a gamble that there would be an audience for this stuff.

I remember watching Totoro as a child not knowing it was an anime. Later I watched The Guyver (1989 OVA) on VHS at my cousin’s. Evangelion and Lodoss War were the first fansubs I watched knowing they were Japanese.

Chobits